If the cost function is C(x) = 5x^2 + 20x + 100, find the minimum cost. (2020)

If the cost function is C(x) = 5x^2 + 20x + 100, find the minimum cost. (2020)

Step 1: Identify the cost function, which is C(x) = 5x^2 + 20x + 100.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = 5, b = 20, and c = 100.
Step 3: To find the minimum cost, use the formula for the x-coordinate of the vertex of a parabola, which is x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -20/(2*5).
Step 5: Calculate the denominator: 2*5 = 10.
Step 6: Now calculate x: x = -20/10 = -2.
Step 7: To find the minimum cost, substitute x = -2 back into the cost function: C(-2) = 5(-2)^2 + 20(-2) + 100.
Step 8: Calculate (-2)^2 = 4, so C(-2) = 5*4 + 20*(-2) + 100.
Step 9: Now calculate 5*4 = 20 and 20*(-2) = -40.
Step 10: Combine these results: C(-2) = 20 - 40 + 100.
Step 11: Finally, calculate 20 - 40 = -20, so C(-2) = -20 + 100 = 80.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find their minimum or maximum values using the vertex formula.
Cost Functions – Applying cost functions in economics to determine the minimum cost associated with production levels.